# Properties

 Label 578.a Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 578.a

Label Equation
578.a.2312.1 $$y^2 + (x^2 + x)y = x^5 - 2x^4 + 2x^3 - 2x^2 + x$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$( 1 - T )( 1 + T + 2 T^{2} )$$
$$17$$$$( 1 - T )( 1 + T )$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$$
$$5$$$$( 1 + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$
$$7$$$$( 1 - 4 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$( 1 - 6 T + 11 T^{2} )( 1 + 11 T^{2} )$$
$$13$$$$( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$19$$$$( 1 + 4 T + 19 T^{2} )^{2}$$
$$23$$$$( 1 - 4 T + 23 T^{2} )( 1 + 23 T^{2} )$$
$$29$$$$( 1 - 6 T + 29 T^{2} )( 1 + 29 T^{2} )$$
$\cdots$$\cdots$

$$\mathrm{ST} =$$ $G_{3,3}$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)$$
Of $$\GL_2$$-type over $$\Q$$
All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.